Everyone makes good points but forgot one other key difference: in CC (at least in A Bridge Too Far which is the last one I got) you can win a battle by parking one hidden soldier right on top of an objective despite the fact that the rest of your forces have been routed, or that you can spend half an hour wasting your time and ammo hunting for one last half squad of the enemy that won't surrender!
Don't get me wrong, I loved CC at the time. Its been surrpassed by CM IMHO and if you take the time to learn it you'll just be learning less realistic tactics. Of course I play these games as a competative/interactive/play-with-little-toy-vehicles/ultimately-historical-learning-excersizes, so your interests may guide you to different conclusions.